The value of `int_(-pi//2)^(pi//2)(x^(2)+x cosx+tan^(5)x+1)dx` is equal to

Options

A. 0
B. 2
C. `pi`
D. none of these

Answer

To solve the integral \( I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( x^2 + x \cos x + \tan^5 x + 1 \right) dx \), we will analyze each term in the integrand to determine if they are even or odd functions.

### Step 1: Split the Integral
We can express the integral as:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^2 \, dx + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x \cos x \, dx + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \tan^5 x \, dx + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 1 \, dx \]

### Step 2: Identify Even and Odd Functions
1. **For \( x^2 \)**:
- \( f(-x) = (-x)^2 = x^2 = f(x) \) (even function)

2. **For \( x \cos x \)**:
- \( f(-x) = -x \cos(-x) = -x \cos x = -f(x) \) (odd function)

3. **For \( \tan^5 x \)**:
- \( f(-x) = \tan(-x)^5 = (-\tan x)^5 = -\tan^5 x = -f(x) \) (odd function)

4. **For \( 1 \)**:
- \( f(-x) = 1 = f(x) \) (even function)

### Step 3: Apply Properties of Integrals
Using the properties of integrals:
- The integral of an odd function over a symmetric interval around zero is zero.
- The integral of an even function can be simplified as:
\[ \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx \]

Thus, we have:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^2 \, dx + 0 + 0 + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 1 \, dx \]
This simplifies to:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^2 \, dx + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 1 \, dx \]

### Step 4: Calculate the Integrals
1. **Calculate \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^2 \, dx \)**:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^2 \, dx = 2 \int_{0}^{\frac{\pi}{2}} x^2 \, dx = 2 \left[ \frac{x^3}{3} \right]_{0}^{\frac{\pi}{2}} = 2 \left( \frac{(\frac{\pi}{2})^3}{3} - 0 \right) = 2 \cdot \frac{\pi^3}{24} = \frac{\pi^3}{12} \]

2. **Calculate \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 1 \, dx \)**:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 1 \, dx = 2 \int_{0}^{\frac{\pi}{2}} 1 \, dx = 2 \left[ x \right]_{0}^{\frac{\pi}{2}} = 2 \cdot \frac{\pi}{2} = \pi \]

### Step 5: Combine Results
Now, combine the results:
\[ I = \frac{\pi^3}{12} + \pi \]

### Final Answer
Thus, the value of the integral is:
\[ I = \frac{\pi^3}{12} + \pi \]